Book of Proof - Amazon S3

153
S 1 : 1 = 1 2
S 2 : 1 + 3 = 2 2
S 3 : 1 + 3 + 5 = 3 2
.
.
S n : 1 + 3 + 5 + 7 + ··· + (2n − 1) = n 2
.
.
Our question is: Are all of these statements true?
Mathematical induction is designed to answer just this kind of question.
It is used when we have a set of statements S 1 , S 2 , S 3 ,..., S n ,..., and we
need to prove that they are all true. The method is really quite simple.
To visualize it, think of the statements as dominoes, lined up in a row.
Imagine you can prove the first statement S 1 , and symbolize this as
domino S 1 being knocked down. Additionally, imagine that you can prove
that any statement S k being true (falling) forces the next statement S k+1
to be true (to fall). Then S 1 falls, and knocks down S 2 . Next S 2 falls and
knocks down S 3 , then S 3 knocks down S 4 , and so on. The inescapable
conclusion is that all the statements are knocked down (proved true).
The Simple Idea Behind Mathematical Induction
S 1 S 2 S 3 S 4 S 5 S 6 ··· S k S k+1 S k+2 S k+3 S k+4 ···
Statements are lined up like dominoes.
S 2 S 3 S 4 S 5 S 6 ··· S k S k+1 S k+2 S k+3 S k+4 ···
(1) Suppose the first statement falls (i.e. is proved true);
···
S k+2 S k+3 S k+4 ···
S 1
S k
S k+1
S1
S2
S3
S4
S5
S6
(2) Suppose the k th falling always causes the (k + 1) th to fall;
S1
S2
S3
S4
S5
S6
···
Sk
S k+1
S k+2
S k+3
···
Then all must fall (i.e. all statements are proved true).